Theorem idlcl 38046

Description: An element of an ideal is an element of the ring. (Contributed by Jeff Madsen, 19-Jun-2010.)

Hypotheses

Ref	Expression
idlss.1	⊢ 𝐺 = (1st ‘𝑅)
idlss.2	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
idlcl	⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ 𝑋)

Proof of Theorem idlcl

Step	Hyp	Ref	Expression
1		idlss.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
2		idlss.2	. . 3 ⊢ 𝑋 = ran 𝐺
3	1, 2	idlss 38045	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋)
4	3	sselda 3963	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ran crn 5660  ‘cfv 6536  1^st c1st 7991  RingOpscrngo 37923  Idlcidl 38036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6538  df-fv 6544  df-ov 7413  df-idl 38039

This theorem is referenced by:  idlsubcl  38052